Editing Graphic matroid (section)

==Representation==
The graphic matroid of a graph <math>G</math> can be defined as the column matroid of any [[incidence matrix|oriented incidence matrix]] of <math>G</math>. Such a matrix has one row for each vertex, and one column for each edge. The column for edge <math>e</math> has <math>+1</math> in the row for one endpoint, <math>-1</math> in the row for the other endpoint, and <math>0</math> elsewhere; the choice of which endpoint to give which sign is arbitrary. The column matroid of this matrix has as its independent sets the linearly independent subsets of columns.

If a set of edges contains a cycle, then the corresponding columns (multiplied by <math>-1</math> if necessary to reorient the edges consistently around the cycle) sum to zero, and is not independent. Conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is independent. Therefore, the column matrix is isomorphic to <math>M(G)</math>.

This method of representing graphic matroids works regardless of the [[field (mathematics)|field]] over which the incidence is defined. Therefore, graphic matroids form a subset of the [[regular matroid]]s, matroids that have [[Matroid representation|representations]] over all possible fields.<ref name="tutte65"/>

The lattice of flats of a graphic matroid can also be realized as the lattice of a [[Arrangement of hyperplanes|hyperplane arrangement]], in fact as a subset of the [[Braid group|braid arrangement]], whose hyperplanes are the diagonals <math>H_{ij}=\{(x_1,\ldots,x_n) \in \mathbb{R}^n \mid x_i = x_j\}</math>. Namely, if the vertices of <math>G</math> are <math>v_1,\ldots,v_n,</math> we include the hyperplane <math>H_{ij}</math> whenever <math>e = v_iv_j</math> is an edge of <math>G</math>.